Making wheels spin as car moves on local Y axis
You cannot easily do this, because it is impossible. @Nathan is right, but I'll add here my explanation:

When moving the car in the scene, you want the wheels to correctly follow - let's move the car 2 different ways:



In first example the wheels traveled longer distance, yet the end car position is the same.

Without Blender accessing and knowing what the path was, it is impossible to compute the wheels rotation, because it can be anything based on how the car was manipulated. When you interactively move objects in the viewport, Blender does not store your moves. The bones only remember their rest and their final (pose) position.

Blender can access what the path was from:


animation keyframes
from defined path the car is traveling (follow path constraint)
from simulation that treats the movement on frame-by-frame basis (you can have a little python script running in background as app.handler and turning the wheels, it could look similar to this, or you can implement this with nodes in Animation Nodes addon)


The easiest and most common method would be to use the path. How to setup a driver that rotates the wheel is here:

How can we animate a ball rolling over a path?

It's convenient to offset the curve for left and right wheels and setup both sides separately, so they rotate correctly when turning.